Born in Barlad, her parents were Francisc Paximade, who came from Tenedos and established a cereal export business at Galati, and Anastasia (nee Thomaide); the two married in 1873. On her father's side, she descended from an old and noble Greek family; through her mother, she was of Greek, Turkish and Romanian origin. Orphaned at a young age, she was raised in the home of Junimist George Panu and briefly with Ion Luca Caragiale, through whom she came to know Alexandru Vlahuta and other contemporary writers. Between 1884 and 1890, she had an incomplete education at the Varlaam and Drouhet boarding schools in her native town. Through her husband, the bank clerk Francisc Farago, she was drawn into socialist circles, attending lectures by Ioan Nadejde, Constantin Dobrogeanu-Gherea, Vasile Mortun and Constantin Stere, also meeting Barbu Branisteanu and Christian Rakovsky.
She made her debut with the 1899 poem "Gandul truditilor", which appeared in Romania muncitoare. Her work also appeared in Adevarul, Epoca, Samanatorul, Floarea darurilor, Revista noastra, Fat-Frumos, Luceafarul and Ramuri. She sometimes signed with the pen names Fatma, Andaluza and Ellen. From 1921 until her death, she headed the Alexandru and Aurelia Aman foundation in Craiova, which today is the Elena Farago Memorial House. She also worked as an inspector for children's charity homes. Farago founded Nazuinta magazine in 1922.
Her first book was Versuri (1906), followed by poetry volumes Soapte din umbra (1908), Soaptele amurgului (1920) and Nu mi-am plecat genunchii (1926); these reappeared in editions she put together (Poezii alese, 1924; Poezii. 1906-1926, 1928; Poezii, 1937), or posthumously, in partly re-edited versions (Poezii, 1957; Versuri, 1978). Farago also published poems for children (Pentru copii, vol. I, 1913, vol. II, 1920; Copiilor, 1913; Din traista lui Mos Craciun, 1920; Bobocica, 1921; Sa nu plangem, 1921; A ciocnit cu ou de lemn, 1943; Intr-o noapte de Craciun, 1944; 4 gaste nazdravane, 1944; Plugusorul jucariilor, 1944), and children's prose books (Sa fim buni, 1922; Ziarul unui motan, 1924; Intr-un cuib de randunica, 1926; Sa nu minti, sa nu furi, 1944).
She translated from Henrik Ibsen, Friedrich Nietzsche, Catulle Mendes, French classical and symbolist poetry (Emile Verhaeren, Henri de Regnier, Paul Verlaine, Sully Prudhomme, Edmond Haraucourt, Maurice Maeterlinck), Anatole France and Lafcadio Hearn. Some of these texts appeared only in magazines, while others were published in the Biblioteca pentru toti and Lectura series. She won prizes from the Romanian Academy (1909, 1921), the Femina Prize (1925) and the national prize for poetry (1937). An accomplished poet of chaste love, delicate confessions and motherhood, Farago is also among the more prominent names in Romanian children's literature. She died in Craiova. Her daughter Coca also became a writer.